262
[387
When XI > 24 and the curve does not meet the axis at all, the constant in (3) must be retained, and the difficulty is much increased. If we suppose that dy/ds = + 1 when x — az and dy/ds = — 1 when x = al, we can determine p0 as well as the constant of integration, and (3) becomes
.(20)
mi eror , „ "fa^W^ '
Cto Cj-L l-t-o ---- L<fl
We may imagine a curve to be traced by means of this equation. We start from the point A where y — 0, as — a^ and in the direction perpendicular to OA, and (as before) we are told in what direction to proceed at any point reached. When x — alt the tangent must again be parallel to the axis, but there is nothing to ensure that this occurs when y = 0. To secure this end and so obtain an annular form of equilibrium, a-o^jT must be chosen suitably, but there is no means apparent of doing this beforehand. The process of curve tracing can only be tentative.
If we form the expression for the curvature as before, we obtain
-x ......(21)
by means of which the curves may be traced tentatively.
If we retain the normal PQ, as we may conveniently do in using Boys' method, we have the simpler expression
2
__ _ .— ------ I Vrtv __ n '•* __ n v\ J_ _—
PQ
aa2) +
.(22)
When the radius CP of the section is very small in comparison with the radius of the ring OC, the conditions are approximately satisfied by a circular
Kg. 2.
form. We write GP = r, OG - a, PGA = #. Then, r being supposed constant, the principal radii of curvature are r and a sec 6 + r, so that the equation of equilibrium is
Po 1 cos# ta2 .emi-transparent celluloid. At one point on the scale a fine pencil point protrudes through a small hole and describes the diminutive circular arc. Another point of the scale at the required distance occupies the centre of the circle and is held temporarily at rest with the aid of a small brass tripod standing on sharp needle points. After each step the celluloid is held firmly to the paper and the tripod is moved to the point of the scale required to give the next value of the curvature. The ordinates of the curve so drawn are given in the second and fifth columns of the annexed table. It will be seen that from x — 0 to x = 2 the curve is very flat. Fig. (1).